FIZZA SARFARAZ ABBOTTABAD UNIVERSITY OF SCIENCE & TECHNOLOGY (A.U.S.T)

CORRELATION

HISTORYGALTON:Obsessed with measurement Tried to measure everything from the weather to female beauty Invented correlation and regression KARL PEARSONformalized Galton's methodinvented method

CORRELATION

Measure of the degree to which any two variables vary together.orSimultaneously variation of variables in some direction.e.g., iron bar

INTRODUCTIONAssociation b/w two variablesNature & strength of

relationship b/w two variablesBoth random variablesLies b/w +1 & -10 = No relationship b/w

variables-1 = Perfect Negative

correlation+1 = Perfect positive

correlation

Ice-cream - Temperature

MATHEMATICALLY

ny)(

y.nx)(

x

nyx

xyr

22

22

EXAMPLEAnxiety Anxiety

(X)(X)Test Test

score (Y)score (Y)XX22 YY22 XYXY

1010 22 100100 44 202088 33 6464 99 242422 99 44 8181 181811 77 11 4949 7755 66 2525 3636 303066 55 3636 2525 3030

∑∑X = 32X = 32 ∑∑Y = 32Y = 32 ∑∑XX22 = 230 = 230 ∑∑YY22 = 204 = 204 ∑∑XY=12XY=1299

Calculating Correlation Calculating Correlation CoefficientCoefficient

94.)200)(356(

102477432)204(632)230(6

)32)(32()129)(6(22

r

r = - 0.94

Indirect strong correlation

METHODS OF STUDYING CORRELATION

METHODS

SCATTER DIAGRAM

KARL PEARSON COEFFICIENT

SPEARMAN'S RANK

SCATTER DIAGRAM

Rectangular coordinate

Two quantitative variables

1 variable: independent (X) & 2nd:

dependent (Y)

Points are not joined

KARL PEARSON COEFFICIENT

• Statistic showing the degree of relationship b/w two variables.

• represented by 'r'• called Pearson's correlation

SPEARMAN'S RANKActual measurement of objects/individuals Actual measurement of objects/individuals

not availablenot availableAccurate assesment is not possibleAccurate assesment is not possibleArranged in orderArranged in orderOrdered arrangement: RankingOrdered arrangement: RankingOrder Given to object: RanksOrder Given to object: RanksCorrelation blw two sets X & Y: Rank Correlation blw two sets X & Y: Rank

correlationcorrelation

PROCEDURE

Rank values of X from 1-n Rank values of X from 1-n n: # of pairs of values of X & Yn: # of pairs of values of X & Y Rank Y from 1-nRank Y from 1-n Compute value of 'di' by Xi - YiCompute value of 'di' by Xi - Yi Square each di & compute ∑diSquare each di & compute ∑di22

Apply formula; Apply formula; 2

s 2

6 (di)r 1

n(n 1)

EXAMPLE In a study of the relationship between level education and In a study of the relationship between level education and

income the following data was obtained. Find the relationship income the following data was obtained. Find the relationship between them and comment.between them and comment.

samplenumbers

level education(X)

Income(Y)

A Preparatory.Preparatory. 25B Primary.Primary. 10C University.University. 8D secondarysecondary 10E secondarysecondary 15F illitilliterateerate 50G University.University. 60

X Y rankX

rankY

di di2

A Preparatory

25 5 3 2 4

B Primary 10 6 5.5 0.5 0.25

C University 8 1.5 7 -5.5 30.25

D secondary 10 3.5 5.5 -2 4

E secondary 15 3.5 4 -0.5 0.25

F illiterate 50 7 2 5 25G university 60 1.5 1 0.5 0.25

∑ di2=64

Conclusion:Conclusion:There is an indirect weak correlation There is an indirect weak correlation

between level of education and income.between level of education and income.

1.0)48(7

6461

sr

TYPES

Types

Type 1 Type 2 Type 3

TYPE 1

Type 1

Negative NO Perfect

Positive

• POSITIVE - both either increase or decrease

• NEGATIVE - one increase while other decrease

• NO - no correlation• PERFECT - both

variables are independents

EXAMPLES

+ive Relationships• WAter consumption

& temperature• Study times &

grades

-ive Relationships• Alcohol

consumption & driving ability

• Price & Quantity demanded

TYPE 2

Type 2

Linear

Non-linear

• LINEAR - Perfect straight line on graph

• NON-LINEAR - Not a perfect straight line

TYPE 3

Type 3

Simple Multiple Partial

• SIMPLE - 1 independent & 1 dependent variable

• MULTIPLE - 1 dep & more than 1 indep variable

• PARTIAL - 1 dep & more than 1 indep variable bt only 1 indep variable is considered while other const

COEFFICIENT OF CORRELATION

Measure of the strength of linear relationship b/w two variables.Represented by 'r''r' lies b/w +1 & -1-1 ≤ r ≤ +1 +ive sign = +ive linear correlation -ive sign = -ive linear correlation

MATHEMATICALLY 'r'

2 2 2 2

( )( )

[ ( ) ][ ( ) ]

n xy x yr

n x x n y y

INTERPRETATION OF 'r'

INTERPRETATION-1 ≤ r ≤ +1

-1 10-0.25-0.75 0.750.25

strong strongintermediate intermediateweak weak

no relation

perfect correlation

perfect correlation

Directindirect

CORRELATIION: LINEAR RELATIONSHIPS

0

20

40

60

80

100

120

140

160

180

0 50 100 150 200 250

Drug A (dose in mg)

Sym

ptom

Inde

x

0

20

40

60

80

100

120

140

160

0 50 100 150 200 250

Drug B (dose in mg)

Sym

ptom

Inde

x

Srong Relationship → Good linear fitPoints clustered closely around a line show a

strong correlation. The line is a good predictor (good fit) with the data. The more spread out the points, the weaker the correlation, and the less good the fit. The line is a REGRESSSION line (Y = bX + a)

r : shows relationship b/w variables either +ive or -ive

r2 : shows % of variation by best fit line

Example:Example:

A sample of 6 children was selected, data about their age A sample of 6 children was selected, data about their age in years and weight in kilograms was recorded as shown in years and weight in kilograms was recorded as shown in the following table . It is required to find the in the following table . It is required to find the correlation between age and weight.correlation between age and weight.

serial # Age X I.V(years)

Weight Y D.V(Kg)

1 7 122 6 83 8 124 5 105 6 116 9 13

Serial n.

Age (years)

(x)

Weight (Kg)(y)

xy X2 Y2

1 7 12 84 49 1442 6 8 48 36 643 8 12 96 64 1444 5 10 50 25 1005 6 11 66 36 1216 9 13 117 81 169

Total ∑x=41

∑y=66

∑xy= 461

∑x2=291

∑y2=742

r = 0.759r = 0.759strong direct correlation strong direct correlation

2 2

41 664616r

(41) (66)291 . 7426 6

2 22 2

x yxy

nr( x) ( y)

x . yn n

APPLICATIONS

Estimating & improving.,Seasonal sales for departmental storesQuantity demanded & productionMotivating tools for employeesCost of products demandedaccuracy of estimations for demands for sailsInflation & real wageOil exploration

Moreover, Radar system is field where correlation is vehicle to map distance&in communication, for instance in digital receivers.

SPSS TUTORIAL

1.Analyz2.Correlate 3.(Bivariate)

Points to be noted:Confidence LevelCorrelation is highly significant 0.01**Correlation is significant 0.05*